Basepoints in Khovanov homology and nonorientable surfaces

Abstract

We enhance the Khovanov TQFT using basepoint actions, over the field with two elements. Our enhanced Khovanov TQFT behaves similarly to gauge/Floer theoretic invariants of the double branched cover with opposite orientation: they both are invariant, in a certain sense, under taking the connected sum with the standard $\mathbb{RP}^{2}$ with Euler number -2, and they both vanish after taking the connected sum with the standard $\mathbb{RP}^{2}$ with Euler number 2. This invariance property answers a version of a question posed by Lipshitz and Sarkar. Furthermore, our construction establishes, as a special case, functoriality for the pointed Khovanov homology defined by Baldwin, Levine, and Sarkar.